Elliott attended Malvern College and trained at the Royal Academy of Dramatic Art in London. He was asked to leave the academy after one term. As Elliott later recalled, They wrote to my mother and said, "Much as we like the little fellow, he's wasting your money and our time. Take him away!"[6]

Despite being described by British Film Institute's Screenonline as an actor of "versatile understanding and immaculate technique",[1] Elliott described himself as an instinctive actor and was a critic of Stanislavski's system of acting, saying: "I mistrust and am rather bored with actors who are of the Stanislavski school who think about detail."[13]

Privately bisexual,[14] Elliott was married twice: first to actress Virginia McKenna for a few months in 1954, and later, in an open marriage, to American actress Susan Robinson (7 March 1942 – 12 April 2007), with whom he had two children, Mark and Jennifer. Jennifer committed suicide by hanging in 2003.[14]

Elliott was diagnosed with HIV in 1987[14] and died of AIDS-related tuberculosis at his home in Santa Eulària des Riu on Ibiza, Spain, on 6 October 1992 at the age of 70. Tributes were paid by actors Sir Donald Sinden and Sir Peter Ustinov, playwright Dennis Potter, and former wife Virginia McKenna. Sinden said, "He was one of the finest screen actors and a very special actor at that. He was one of the last stars who was a real gentleman. It is a very sad loss." Ustinov said, "He was a wonderful actor and a very good friend on the occasions that life brought us together." Potter commented, "He was a complicated, sensitive and slightly disturbing actor. Not only was he a very accomplished actor, he was a dry, witty and slightly menacing individual. As a man, I always found him very open, very straightforward and very much to the point." McKenna added: "It is absolutely dreadful, but the person I am thinking of at the moment more than anybody is his wife. It must be terrible for her."[15]Ismail Merchant described Elliott as "an all-giving person, full of life ... He had an affection and feeling for other actors, which is very unusual in our business."[16]